1. Approximation of the Propagator for the Harmonic Oscillator
By Logan Thomas
Advisor: Dana Fine

2. Abstract
  The propagator, which is a function of an initial position, a final position and a time, completely determines the quantum mechanics of a point particle in a given potential. The approximate propagators are explicitly calculated for the case of the harmonic oscillator potential. This project presents numerical investigation of the accuracy of certain approximate propagators for various values of N and t. This is feasible for this case, as the exact propagator is known.

3. The Problem
The propagator gives a probability amplitude of a point particle moving to a new place at a given time.
The actual Propagator for a 1 dimensional particle is known
The experimental first order Propagator was derived by N. Makri and W. Miller

4. Problem cont.
My Job:
Test the accuracy of the first order propagator approximation compared to the actual propagator.

5. Task 1: Simplification
Actually proved to be very hard, figuring out how to put these two equations into MATLAB.

6. Simplification (cont.)
As specified in the paper the proposed approximation is valid when:
My final equations after the simplifications were:

7. Task 2: Matlab Running an error analyses:
((experimental – actual)/actual)*100
The paper showed their error plot as being:

8. Current State
At this current moment, My MATLAB program, is producing this graph for the error:
Obviously there is an error in my MATLAB program somewhere.

9. Future Plans
Iron out whatever problems/figure out the source of the problem in MATLAB
Find the actual % error of their approximation as N gets larger.
Attempt to make the approximations better.
Also try for different values of x.

10. References
Nancy Makri, William H Miller, Correct short term propagator for feynman Path integration by power series expansion in $/delta t$. Department of Chemistry, University of California. (1988)